A brine solution containing 3kg of salt per gallon flows at a constant rate of 20gal/minute into a tank that holds 400 gallons of pure water. The mixture in the tank is kept well stirred and flows out of the tank at the rate of 25gal/min.
a) find the mass of salt in the tank at time t
b) find the concentration of salt in the tank when there is 200 gallons of solution
Great question! So the problem states that the tank starts with 400 gals, so that is where the 400 comes from. Next, you have 20 gals flowing in and 25 flowing out, per minute, so the total volume at any point in time is 400-5t. This is important in that term, because the amount of salt flowing out depends on the concentration in the tank at any given time point!
You have a separable differential equation. You introduce the exponential because you must move S to the left hand side, in the denominator, where the integral of 1/S is ln(S), and the right hand side is straightforward integration.